1. **State the problem:** Differentiate the function $y = \sin^2(x^2)$ with respect to $x$. 2. **Recall the formula:** To differentiate a composite function like $\sin^2(u)$ where $u = x^2$, use the chain rule: $$\frac{d}{dx}[\sin^2(u)] = 2\sin(u) \cdot \cos(u) \cdot \frac{du}{dx}$$ 3. **Identify inner functions:** Here, $u = x^2$, so $$\frac{du}{dx} = 2x$$ 4. **Apply the chain rule:** Substitute $u$ and its derivative into the formula: $$\frac{dy}{dx} = 2 \sin(x^2) \cdot \cos(x^2) \cdot 2x = 4x \sin(x^2) \cos(x^2)$$ 5. **Simplify using a trigonometric identity:** Recall that $$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$ so $$4x \sin(x^2) \cos(x^2) = 2x \cdot 2 \sin(x^2) \cos(x^2) = 2x \sin(2x^2)$$ **Final answer:** $$\frac{dy}{dx} = 2x \sin(2x^2)$$